Course MATH 2121 - Mathematical Modeling and Simulation - Spring 2025

Official Information
Course Number:	MATH 2121.001
Course Title:	Mathematical Modeling and Simulation
Times:	TR 11:00-12:20
Room:	010 Wachman Hall

Instructor:	Benjamin Seibold
Instructor Email:	seibold(at)temple.edu
Instructor Office:	518 Wachman Hall
Instructor Office Hours:	TR 12:20-1:30

Official:	Course Syllabus

Course Materials:	There is no required textbook for this course. All materials will be provided by the instructor (including lecture notes, short videos, software) or acquired via assigned reading. For students who wish to read more on the topic of modeling and simulation in general (not directly related to the present course), the following textbooks provide suitable sources: Marco A. Janssen, Introduction to Agent-Based Modeling, website, 2012 Steven F. Railsback, Volker Grimm, Agent-based and Individual-based Modeling: A Practical Introduction, Princeton University Press, 2011.
Prerequisites:	Programming: MATH1034/CIS1051/CIS1057/CIS1068 or equivalent. Mathematics: MATH1042 or equivalent. See Course Catalog for up-to-date prerequisites.
Course Goals:	By the end of the semester, students will be able to build models and simulations for complex emergent phenomena, and to systematically analyze and interpret the simulation results.
Topics Covered:	This course introduces the concept of (a) building a mathematical model of a real-world process, (b) using computational resources to simulate the model, and (c) properly interpreting the results. The main focus lies on processes with many interacting agents, such as: traffic flow, spread of diseases, animal swarming, economic markets, social networks, robotics. The course provides an overview of model building concepts, training on the implementation of models in a computing environment, as well as theoretical background on how to analyze and understand large-scale emergent structures (such as traffic waves, stock market crashes, swarm intelligence, etc.). General interest in mathematical model building and in programming is required.
Course Grading:	Homework and in-class activities: 30%; course project: 45%; examination: 25%.
Final Exam Date:	05/06/2025.
Attendance Policy:	Students are expected to attend every class. If a student cannot attend a class for some justifiable reason, they are expected to contact the instructor before class. Students are expected to actively participate in the in-class activities and to be properly prepared to do so, including having completed any pre-class assignments and providing suitable hardware and software, as explained during the first class.
Homework:	Homework problem sets will be assigned every 1.5 to 2 weeks, and must be submitted by the submission date to the course instructor (any extension must be requested and granted before the submission deadline), involving both a programming part and a written/typed part as instructed on the problem set. All submitted work must be the student's own work, though discussion with classmates is allowed and even encouraged. All submitted software must adhere to best practices, including a description of purpose, clear comments, and properly citing code segments from other sources.
Programming:	Computational problems will be conducted using MATLAB. Temple University has a site license for MATLAB, and it can be installed free-of-charge on personal computers by the steps given on the ITS website. For the course project, other languages may also be used, but for specific assignments the usage of MATLAB is required.
Course Project:	Over the course of the semester, each student will work on an individual course project related to the course material, on a topic found in discussion with the course instructor. Each course project should demonstrate the major themes of model building, computer simulation, presentation, interpretation, and analysis of results. Project topics suggested by the student are very welcome, and the instructor will make an effort to admit topics of specific interest to students. Each project involves a project proposal, a midterm report, a final report, and a project presentation (due dates see below).
Course Schedule (Plan)
01/14/2025 Class 1	Principles of agent-based models, Mexican wave example, Matlab basics Read: Agent-based model, Collective animal behavior, Wave (audience), The first(?) wave, Full stadium wave
01/16/2025 Class 2	Deterministic vs. stochastic models, contagion model, Matlab data structures and randomness Read: Infection, Stochastic
01/21/2025 Class 3	Monte-Carlo method, nested functions in Matlab Read: Monte Carlo method
01/23/2025 Class 4	Random walks, randomness in Matlab Read: Random walk
01/28/2025 Class 5	Random walks in 2d, histograms Read: Histogram, Lévy flight
01/30/2025 Class 6	Random walks with bias, simulated annealing Read: Simulated annealing
02/04/2025 Class 7	Butterfly movement model Read: Hill-topping
02/06/2025 Class 8	Run-and-tumble, E. coli model, chemotaxis Read: Chemotaxis
02/11/2025 Class 9	Agents affecting field quantities
02/13/2025 Class 10	Diffusion (macroscopic and microscopic) Read: Fick's law, Finite volume method, Diffusion
02/18/2025 Class 11	Interactions between agents, population dynamics models Read: Population dynamics
02/20/2025 Class 12	Theory of population dynamics Read: Logistic growth, Predator-prey equations, Competition, Mutualism
02/25/2025 Class 13	Dynamical systems, equilibria, stability Read: Linear dynamical system, Equilibrium point, Stability theory, Saddle, Attractor
02/27/2025 Class 14	Numerical time stepping Read: Ordinary differential equation, Euler method
03/11/2025 Class 15	Runge-Kutta methods, SIR model Read: Runge-Kutta methods, SIR model
03/13/2025 Class 16	Traffic flow: car following models, traffic waves, instabilities Read: Traffic flow, Microscopic traffic flow model, Traffic wave
03/18/2025 Class 17	Traffic flow: cellular automaton models Read: Nagel-Schreckenberg model
03/20/2025 Class 18	Conway's game of life Read: Game of life, LifeWiki, GOL glossary
03/25/2025 Class 19	General cellular automaton models Read: Cellular automaton, Rule 184
03/27/2025 Class 20	Models for crowd dynamics Read: Crowd
04/01/2025 Class 21	Swarming and flocking Read: Swarm behaviour, Flocking, Bait ball
04/03/2025 Class 22	Ant colony simulation Read: Ant colony, How ants find food, Ant colony optimization
04/08/2025 Class 23	Economic models: fundamentals and trading Read: Supply and demand, Money, Commodity, Free market, Trade
04/10/2025 Class 24	Economic models: stock market Read: Stock, Stock valuation, Share price, Efficient-market hypothesis, Market trend
04/15/2025 Class 25	Opinion dynamics Read: Opinion
04/17/2025 Class 26	Calibration, verification, validation Read: Calibration, Verification, Validation
04/22/2025 Class 27	Project presentations
04/24/2025 Class 28	Project presentations
05/06/2025	Final Examination
Course Reading and Presentation Materials
Class 1: Introduction to Agent-Based Models Class 3: Stochasticity and Monte-Carlo Method Class 4: Random Walks Class 5: Random Walks, Levy Flight, and Simulated Annealing Class 7: Butterfly Hilltopping Class 8: Butterfly Corridor and Chemotaxis Class 10: Kernel Density Estimation Class 10: Diffusion Class 12: Population Dynamics Class 13: Phase Portraits Class 14: Runge-Kutta Methods Class 15: ODE Approximation and SIR Model Class 16: Car Following Models
Instructional Videos
Traffic Flow: Car-Following Models (by Prof. Benjamin Seibold) Nagel-Schreckenberg Model (by Prof. Rujeko Chinomona) Cellular Automata (by Prof. Rujeko Chinomona) Game of Life (by Prof. Rujeko Chinomona) Swarming and Flocking (by Prof. Benjamin Seibold) Ant Colony Simulation (by Prof. Benjamin Seibold) Economy Supply and Demand (by Prof. Benjamin Seibold) Stock Market (by Prof. Rujeko Chinomona) Opinion Dynamics (by Prof. Rujeko Chinomona)
Matlab Programs
Simple model for a mexican wave in a sports stadium, generated by spectators who stand up and sit down to create the wave as an emergent structure: temple_abm_mexican_wave.m Same model as above, but implemented in an alternative fashion: plotting is done graphically, and the update rule is implemented in a non-vectorized fashion: temple_abm_mexican_wave_alternative.m Simple model for a contagion process of agents in a one-dimensional arrangement. Infected agents infect neighbors with a certain probability, and the disease spreads: temple_abm_contagion1d.m Multiple random walkers in one space dimension, whose paths cannot cross (any random step that would cause crossing is rejected). This is a simple example of multiple random walking agents whose paths are not independent: temple_abm_random_walkers.m Random walkers in two space dimensions, exhibiting an animation of their paths. For one walker, each step is drawn uniformly from {-1,0,1}^2; for the other walker, each step is drawn uniformly from [-1,1]^2: temple_abm_random_walk_2d.m Example of a Levy flight (a random walk with heavy-tailed step lengths) in two space dimensions, with log-normally distributed step length (and uniformly distributed angle). The 2d histogram of the final positions is plotted: temple_abm_random_walk_levy_histogram.m Random walk on a given function with a certain probability that an upward step is rejected. The rejection probability increases in time. This is an example of the simulated annealing technique: temple_abm_random_walk_annealing.m Random walk in 2d over a background map, modeling the motion of a butterfly. Each step is either a deterministic uphill motion or a random step: temple_abm_butterfly_animation.m Motion of butterflies via random walk in 2d over a background map. Now, multiple agents are simulated and statistical properties of their paths are extracted (here: the width of corridors): temple_abm_butterfly_corridor_width.m Model for bacteria (such as E. coli) moving in a 2d concentration field. Their motion is run-and-tumble, where the probability to terminate their straight motion depends on whether they "their" concentration value is improving or not: temple_abm_bacteria_run_and_tumble.m Model for bacteria (such as E. coli) moving in a 2d concentration field. In addition to executing a concentration-based run-and-tumble motion, the bacteria consume the concentration in their vicinity: temple_abm_bacteria_run_and_tumble_and_eat.m Constructing density field from point positions in 2d, via histogram and kernel density estimation, using Gaussian kernels: temple_abm_kde.m Diffusion of a 2d concentration field (with no-flux boundary conditions), computed via a macroscopic finite volume method: temple_abm_diffusion_macroscopic.m Comparison of microscopic diffusion (the histogram of many random walkers) with macroscopic diffusion (approximating the diffusion equation via finite volumes): temple_abm_diffusion_micro_vs_macro.m Agents moving in a 2d rectangle, aging (and dying), and producing offspring if exactly two agents are in the same square of the domain: temple_abm_population_migrate_mate_and_age.m Two populations of agents moving in a 2d rectangle, and reproducing and dying. The prey reproduction rate decreases with prey density, and the death rate increases with predator density. The predator reproduction rate increases with prey density, and the death rate increases with predator density: temple_abm_population_predator_prey.m Classical population dynamics without agents. The dynamics of two species are demonstrated in various cases: predator-prey, logistic growth, competition, and mutalism: temple_abm_population_local.m Comparison of three methods for time stepping for ordinary differential equations: temple_abm_ode_time_stepping.m Follow-the-leader traffic model in which drivers equilibrate their car's velocity to that of the vehicle immediately ahead: temple_abm_traffic_follow_the_leader.m A more complex car following model in which drivers have two objective: a) equilibrate their velocity to that of the vehicle immediately ahead, and b) go towards an optimal velocity that depends on the headway: temple_abm_traffic_car_following.m Cellular automaton model for vehicular traffic flow (the Nagel-Schreckenberg model). Vehicles move from cell to cell while constantly adjusting their velocity: temple_abm_traffic_cellular.m Implementation of the popular cellular automaton model game of life by John Conway: temple_abm_cellular_game_of_life.m Same game of life, but now initial configurations are read from data files. temple_abm_cellular_game_of_life_file_ic.m. This program needs the two data files temple_abm_cellular_game_of_life_file_ic_blinker_ship.txt and temple_abm_cellular_game_of_life_file_ic_glider_gun.txt to be in the same directory. Binary cellular model in 1d with immediate neighbor interaction only. This particular example implements rule 184: temple_abm_cellular_rule184.m Cellular model in 2d for the dynamics of crowds. This example has agents that wish to walk to the right, and agents that wish to walk to the left. Each cell can be occupied only by a single agent: temple_abm_cellular_crowd.m Model for swarming birds. Each agent (bird) is moving with a fixed speed, and it adjusts its direction of flight according to other agents nearby. There is a zone of repulsion (radius 1/3) and a zone of alignment (radius between 1/3 and 1): temple_abm_swarming_birds.m Model for foraging ants. Each agent (ant) is moving with a fixed speed, and it adjusts its direction of motion to trace high concentrations of a pheromone field. In turn, every agent leaves a trail of pheromones when walking: temple_abm_swarming_ants.m Model for a market. Producing agents sell goods to consuming agents at a market price. Each agent has a desired price, and a trade occurs when the market price meets or is even better than the agent's desired price. In turn, the market price adjusts according to the balance of willing buyers vs. sellers: temple_abm_economy_supply_and_demand.m Model for a stock market. Agents possess shares of stock, cash, and individual buying and selling prices. The stock price adjusts to balance supply and demand. Whenever possible, trades take place. The agents' prices adjust randomly, plus a momentum component: temple_abm_economy_stock_market.m Opinion dynamics model. In a one-dimensional opinion space, each agent is drawn towards the average opinion of other agents who are nearby in opinion space: temple_abm_opinion_dynamics.m Opinion dynamics model with spatial component. In a two-dimensional space (location, opinion), each agent is drawn towards the average opinion of other agents who are nearby in location as well as in opinion: temple_abm_opinion_dynamics_2d.m
Matlab Introductions and References
The programming in this course is done with MATLAB: Official website and Wikipedia article Introduction to MATLAB - General Information (short) Lab 1 General MATLAB Information (short) Introduction to MATLAB for Engineering Students by David Houcque
Homework Problem Sets
Problem set 1 (due 01/28/2024)
Course Projects
Every student has to conduct an individual course project over the course of the semester. Proposals (written, via email) for projects are due Fri 01/24/2025. Midterm project reports are due Tue 03/11/2025. Final project reports are due Fri 05/02/2025.